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The interaction $-\frac{1}{2}(h{\varphi }^2\Phi +h^{\ast }{\varphi }^{†2}{\Phi }^{†})$ is invariant under a global phase symmetry precisely when the displayed charge relation holds. With $q_{\varphi }=1$, a heavy particle annihilated by $\Phi$ has charge $-2$, so its allowed decay channel is $\Phi \to \bar{\varphi }\bar{\varphi }$. The reverse heavy antiparticle decays into $\varphi \varphi$. Crossing only one heavy external state changes the exchange channel of mixed-species quantum scattering.

James H. Christenson, James Watson Cronin, Val Logsdon Fitch and René Turlay reported the result in 1964. At Brookhaven's AGS, 30 GeV protons struck beryllium; collimators selected a neutral particle beam at 30 degrees. A downstream helium decay volume reduced interactions.^[ref]

The flight distance suppressed the short-lived kaon component. Each arm of a two-arm magnetic spectrometer had spark chambers before and after its magnet. Cherenkov detectors and scintillation counters triggered photographic recording in coincidence.^[ref]

The helium sample yielded $45\pm 9$ signal events; $(2.0\pm 0.4)\times 10^{-3}$ was the ratio to all charged decay channels.^[ref] That denominator must be retained when comparing this result with a total branching fraction.

The evidence was kinematic. Later regenerated-beam quantum interference established that the short- and long-lived states reached the same two-pion final state, excluding alternatives such as distinct pion-like particles.^[ref] The 1964 observation established evidence for CP violation; it did not determine every parameter of the CKM matrix. Inferring time-reversal symmetry violation from CP violation invokes CPT symmetry, rather than a direct comparison of time-reversed transitions in that experiment.

A decay channel specifies a possible final state of a particle decay, such as an electron, a neutrino and an antineutrino in a specified muon decay. Different channels can have different interaction amplitudes, available phase space and experimental visibility.

For disjoint final-state classes, the partial decay widths add: ${\Gamma }_{\mathrm{total}}={\sum }_i{\Gamma }_i$. The branching fraction is ${\Gamma }_i/{\Gamma }_{\mathrm{total}}$, so increasing one partial width changes the fractions assigned to the others even if their own amplitudes are unchanged. Intermediate resonances leading to the same final state can interfere; they cannot always be treated as mutually exclusive counting bins.

Exciton condensation is collective quantum coherence of electron-hole pairs or excitonic degrees of freedom. A finite exciton population alone does not establish it; the central issue is macroscopic phase coherence or the corresponding ordered state.

The physical setting matters. Long-lived excitons in a semiconductor, electron-hole bilayers and driven exciton-polariton systems have different conservation laws, decay channels and interactions. A spectral feature or an increase in emitted intensity therefore needs supporting coherence and dynamical evidence before being interpreted as condensation.^[ref]

The ATLAS experiment and CMS experiment searched proton-proton collisions at the Large Hadron Collider at collision energies of $7$ and $8\mathrm{TeV}$. The ATLAS experiment used silicon tracking, a $2\mathrm{T}$ solenoid, liquid-argon electromagnetic calorimeters and an external muon spectrometer. Its July 2012 paper reported a combined local statistical significance of $6.0$ standard deviations near $126.5\mathrm{GeV}$; this paper result is distinct from the initial 4 July announcement.^[ref]

The CMS experiment used silicon tracking in a $3.8\mathrm{T}$ solenoid, lead-tungstate electromagnetic calorimeters and external muon chambers. Electrons combine tracks and particle showers; muons combine tracking and chamber hits; photons supply shower energies and directions. Selected $H\to \gamma \gamma$ and $H\to Z{Z}^{\ast }\to 4\ell$ candidates, with $\ell =e,\mu$, yielded invariant mass peaks. It checked lepton efficiencies with Z boson decays, estimated fake-lepton backgrounds from control samples with relaxed identification, and fitted the smooth diphoton background from data. Its combined local statistical significance was $5.0$ standard deviations near $125\mathrm{GeV}$.^[ref]

The ATLAS experiment compared signal-plus-background and background-only hypotheses with a profile likelihood, accounting for systematic uncertainties and correlations between decay channels. Diphoton and four-lepton candidates gave precise invariant masses; $W{W}^{\ast }$ decays supplied additional evidence with poorer mass reconstruction because of escaping neutrinos. A local p-value concerns one specified mass; scanning masses requires a look-elsewhere effect correction.^[ref]

Two-photon decay excludes spin one under the usual assumptions, but does not prove spin zero. A mass peak does not determine every Higgs boson coupling. Production rates, branching fractions and angular correlations require additional tests.^[ref]

For a coherently prepared state proportional to $\rho |K_S⟩+|K_L⟩$, evolve each component with ${\lambda }_j=m_j-i{\Gamma }_j/2$. The proper time is $t$, and $\hslash =1$. Since both components reach the same final state, their decay amplitudes add before squaring. Factoring out $|A_S{|}^2$ and a common beam normalization gives the two-pion intensityThe last term is quantum interference. It vanishes upon averaging over an independently random preparation phase, but cannot be discarded from a coherent preparation. Summing the two positive contributions alone would lose the phase-sensitive information.

At late proper time, the short-lived contribution and the quantum interference term die away, leaving the long-lived tail $|{\eta }_{\pi \pi }{|}^2{e}^{-{\Gamma }_{L}t}$. Such a tail is easier to identify after the initially abundant short-lived component has disappeared. The useful experimental balance is between eliminating that component and retaining enough long-lived decays to observe a rare decay channel.

The 1964 report exposed Brookhaven's 80-inch hydrogen bubble chamber to a mass-separated $5.0\mathrm{GeV}/c$ negative-kaon beam. A photographed decay chain, track momenta and bubble density supported the particle identification. Two converted photons helped reconstruct a neutral pion. Alternative assignments were tested kinematically. The authors reported an Omega baryon candidate with $Q=-1$, $S=-3$ and mass $1686\pm 12\mathrm{MeV}/c^2$, reserving a detailed systematic-error discussion for further events.^[ref]

This is an inference from several linked observations. A charged particle can leave a visible bubble-chamber track, while a neutral segment must be reconstructed from its production and decay products. Agreement between possible decay channels, electric charge, momentum conservation and the reconstructed mass is stronger evidence than a suggestive track shape alone. The event tests a predicted hadron pattern; it does not photograph the three strange quarks individually.

For example, an independently constructed two-body kinematic check usesIn a parent rest frame with equal and opposite daughter momenta, the momentum sum vanishes, giving $M{c}^2=E_1+E_2$. In units $c=1$, stipulated daughter masses $m_1=1$, $m_2=1/4$ and momentum magnitude $p=3/4$ give $E_1=5/4$, $E_2=\sqrt{10}/4$ and $M=(5+\sqrt{10})/4\simeq 2.04057$. A common Lorentz boost changes the daughter energies and momenta while preserving this reconstructed invariant mass. These chosen units and numbers are an illustration, not a reanalysis of the 1964 event.

Particle decay is a spontaneous transformation of an unstable particle into allowed final particles. Conservation of energy, momentum, electric charge and the relevant interaction's other conserved quantities constrain the possibilities. A decay can be energetically allowed yet absent or weak because its transition amplitude is suppressed.

For an isolated state with an approximately constant rest-frame decay rate, the surviving population follows an exponential in proper time. The mean lifetime $\tau$ and energy decay width $\Gamma$ obey $\Gamma =\hslash /\tau$. A moving particle's laboratory flight-time distribution also includes time dilation.

Experimentally, detectors reconstruct decay products and sometimes a separated decay vertex. Correcting for acceptance and reconstruction efficiency is necessary before interpreting their counts as a lifetime or branching fraction. Different decay channels compete to make the total width.

Below $m_H=2m_W$, the on-shell two-body decay channel is kinematically closed. An imaginary $p^{\ast }$ in the preceding formula is therefore a sign that this formula no longer applies. Slightly below threshold, the physical process can instead contain an on-shell W boson and an off-shell one, $H\to W{W}^{\ast }\to Wf{\bar{f}}^{\prime }$, followed by decay of the remaining W boson. Its rate requires the virtual-vector propagator, its allowed fermionic decay channels and the corresponding multiparticle Lorentz-invariant phase space. The nonzero decay width of the W boson smooths the idealized threshold. A polarization-resolved rate for two on-shell longitudinal vectors must not be analytically continued and treated as that multiparticle rate.

All three interactions respect color charge; quark flavor is conserved by the strong interaction and the electromagnetic interaction, but can change in the weak interaction. These statements concern charges and net numbers, rather than an unchanging inventory of particles.

A quark transforms in the fundamental representation of the color charge gauge group $SU(3{)}_C$. The strong interaction can change an individual quark's colour by exchanging a gluon, which itself carries color charge. For example, the colour flow $q_r\to q_b+g_{r\bar{b}}$ is allowed: demanding that the outgoing quark remain red would be incorrect. At each interaction vertex the full colour flow is conserved. The colourless photon, W boson, and Z boson instead couple to a quark with an identity factor ${\delta }_{ab}$ in colour space. A weak charged current can therefore change $d_r$ to $u_r$, but cannot turn $d_r$ into $u_b$ without another coloured participant. External isolated hadrons are colour singlets; confinement prevents direct observation of a free coloured state.

For each quark flavor $f$, define the net quark flavor number $F_f=N_f-N_{\bar{f}}$. The strong interaction and electromagnetic interaction preserve every $F_f$: pair creation $g\to s\bar{s}$, for example, changes neither net strangeness nor any other $F_f$. Their quark couplings are diagonal in quark flavor. The weak charged current contains ${\bar{u}}_i{\gamma }^{\mu }(1-{\gamma }^5)V_{ij}{d}_j$, where $V$ is the Cabibbo-Kobayashi-Maskawa matrix, and changes these net numbers. Weak neutral currents are diagonal in tree-level Feynman diagrams; flavor-changing neutral currents can occur through higher-order Feynman diagrams.

The observed hierarchy of mean lifetimes and decay channels supplies evidence: associated strange-particle production through the strong interaction preserves net strangeness, whereas a kaon's weak interaction decay to pions changes it. The electromagnetic interaction process ${\pi }^0\to \gamma \gamma$ has zero net quark flavor before and after annihilation. Neutrino oscillations show that individual lepton family numbers are not exact conserved quantities. Within the usual effectively massless-neutrino approximation, charged current decays such as ${\mu }^{-}\to e^{-}{\bar{\nu }}_e{\nu }_{\mu }$ preserve family numbers only when the emitted neutrinos are counted.

Evidence for three colours includes the hadronic R ratio $R=3{\sum }_f{Q}_f^2$, the colour multiplicities in W boson decay widths, and three-jet events revealing gluon radiation. These test quantum chromodynamics and its colour structure together; they are not direct measurements of a named red or blue charge on a free quark.

The radiative $X^0$ decay conserves strangeness and proceeds electromagnetically. A strong $\Lambda \pi$ decay is closed because even the lighter neutral-pion threshold exceeds the supplied $X^0$ mass. The $Y^{-}$ decay has about $65\mathrm{MeV}$ of available energy, but changes strangeness from $-2$ to $-1$, so it requires the weak interaction. Its lifetime is therefore ordinarily much longer than the electromagnetic lifetime of $X^0$; no numerical lifetime follows from these masses alone.

Both particles are baryons with half-integer spin. A radiative transition to spin $1/2$ can couple through photon multipoles $L\ge 1$, giving $J_X=L\pm 1/2$. Electromagnetic parity is conserved, but without the electric/magnetic multipolarity neither $J_X$ nor its parity is fixed. In the weak $Y^{-}$ decay, coupling orbital $L$ to the final $\Lambda$ spin gives $J_Y=L\pm 1/2$; parity conservation cannot be imposed on that weak amplitude. The supplied modes therefore do not determine unique spin-parity assignments.

The masses and flavours identify the familiar candidates as ${\Sigma }^0$ and ${\Xi }^{-}$. Their established ground-state assignments are both $1/2^{+}$, consistent with the lowest-orbital quark model, but this is additional empirical/model information rather than a consequence of the two decay channels alone.^[ref]

With the conventional particle labels, the allowed cubic decay channels areEach has one tree-level Feynman diagram, consisting of one cubic Feynman vertex. The final identical-leg contractions supply $2!$ and cancel the $1/2!$ in the Lagrangian density.

Taken literally with these labels, the printed $\Phi \to \varphi \varphi$ channel is forbidden: its initial Noether charge is $-2$ and its final Noether charge is $+2$. Its decay amplitude is zero, at every order preserving the global phase symmetry. If the source uses “$\Phi$” to name the heavy antiparticle instead, its intended answer is the nonzero decay amplitude $-h$ for $\bar{\Phi }\to \varphi \varphi$. The following decay width calculation concerns either allowed channel, whose widths are equal by charge conjugation.

For an allowed decay channel in iii, $|\mathcal{M}{|}^2=h^2$. Integrating the spatial Dirac delta distribution in the printed Lorentz-invariant phase-space measure sets ${\mathbf{p}}_2=-{\mathbf{p}}_1$. With $p=|{\mathbf{p}}_1|$ and $E_p=\sqrt{p^2+m^2}$, its remaining value isThe positive root is $p_{\ast }=(M/2)\beta$, where $\beta =\sqrt{1-4m^2/M^2}$, and the Dirac delta distribution contributes the Jacobian determinant $E_{\ast }/(2p_{\ast })$. HenceIt follows that the literal printed integral evaluates toThe printed integration ranges count each final configuration twice, once as $({\mathbf{p}}_1,{\mathbf{p}}_2)$ and once in the opposite order. For two identical final bosons, the physical decay width therefore requires an additional identical final-state symmetry factor $1/2!$:There is one allowed decay channel per heavy particle, not an extra factor of two from summing the two charge-conjugate heavy species. Both expressions vanish below the two-body decay threshold $M=2m$, and the limiting value at threshold is zero. For the literally forbidden labels $\Phi \to \varphi \varphi$, the decay width is instead zero because its decay amplitude vanishes, as explained in iii.

For $g=0$, the invariant scattering amplitude from the tree-level Feynman diagram in i has a pole at $s=M^2$. The intermediate heavy antiparticle can go on shell and live for a time of order its mean lifetime. Near this scattering resonance, inserting a single self-energy is no longer a uniformly small correction. Dyson resummation gives, to leading accuracy in a narrow decay width,Here $M$ is understood as the renormalized resonance mass; the real self-energy shifts the original pole position. The imaginary term removes the unphysical divergence of the fixed-order invariant scattering amplitude.

For equal incoming and outgoing light masses, the two-body momentum ratio cancels in the cross section. The angle-independent invariant scattering amplitude, integrated over the full solid angle with the identical final-state symmetry factor $1/2!$, gives $\sigma =|\mathcal{M}{|}^2/(32\pi s)$. Thus the Breit-Wigner resonance cross section isAt $s\gg M^2$ it reproduces $h^4/(32\pi s^3)$. At the resonance peak,This is also $8\pi /k_{\ast }^2$ for $k_{\ast }=M\beta /2$, the elastic identical-boson S wave partial-wave unitarity limit when this is the only open decay channel. The approximation assumes ${\Gamma }_{\mathrm{phys}}\ll M$ and a resonance sufficiently far from threshold for the width to vary slowly across it.

If one mechanically uses the unsymmetrized ${\Gamma }_{\mathrm{printed}}=2{\Gamma }_{\mathrm{phys}}$ from d, the same displayed Breit-Wigner resonance cross section would have that doubled width and predict a peak $8\pi /(M^2{\beta }^2)$. This is the literal-width substitution, rather than the consistent physical result: the given high-energy cross section already includes the identical-final-particle factor that the printed decay width integral omits.

Counting the mass dimensions of each interaction operator givesIndeed $\Phi (\partial a{)}^2$ and $\Phi F^2$ have dimension five, whereas $\Phi b^2(\partial b{)}^2$ has dimension seven. These are interactions of an effective field theory with dimensionful couplings.

The direct one-vertex decay channels are $\Phi \to aa$, $\Phi \to bbbb$ and $\Phi \to \gamma \gamma$. Each leading scattering amplitude is linear in its corresponding coupling, and a decay width is quadratic in the amplitude. Therefore, holding the parent mass fixed,There are no tree-level interference terms between these different final states.

Every interaction contains an even number of $b$ fields, so the exact discrete symmetry $b\mapsto -b$ forbids $\Phi \to bbb$ to every perturbative order. There is no such prohibition for $\Phi \to bb$.

The latter decay channel is absent at tree level in the displayed interactions, but contracting two $b$ legs at a $\Phi b^2(\partial b{)}^2$ vertex produces a tadpole diagram with two external $b$ legs. For example, contracting the undifferentiated pair leaves the local operator $\Phi (\partial b{)}^2$ multiplied by an integral proportional to $\int d^4\ell /({\ell }^2+i0)$. Contracting the differentiated pair instead gives a local $\Phi b^2$ contribution. Both are allowed effective field theory operators. With a ultraviolet cutoff, these local terms generically carry quadratic or quartic cutoff dependence and require matching coefficients or counterterms.

There is a relevant qualification to a claim that the one-loop decay width must be nonzero. Since $b$ is exactly massless here, the one-vertex tadpole integrals have no external scale and are scaleless integrals. They vanish in dimensional regularization. Thus the correct conclusion is allowed, and generically generated by quantum corrections or matching, not a regulator-independent nonzero one-loop prediction from ${\lambda }_b$ alone. No exact symmetry protects the two-$b$ channel; its numerical rate requires the renormalized coefficients of the allowed operators.

A two-track invariant mass and a direction test answer different questions. In units $c=1$, assign each track the pion mass $m_{\pi }$ and reconstruct its momentum before magnetic deflection. With $p_{\pm }=|{\mathbf{p}}_{\pm }|$ and opening angle $\alpha$,This follows by expanding the squared sum of the two four-momenta. The angle $\theta$ between the reconstructed parent momentum $\mathbf{P}={\mathbf{p}}_{+}+{\mathbf{p}}_{-}$ and the neutral particle beam direction $\hat{\mathbf{n}}$ satisfiesA fully reconstructed two-body decay channel gives the parent mass and $\theta \simeq 0$, within experimental resolution. A three-body decay channel with an unseen neutrino generally misses both constraints; an accidental correct invariant mass alone is insufficient. Notice that the track opening angle $\alpha$ need not be small even when the parent direction angle $\theta$ vanishes.

The 1964 analysis selected $490<M_{\pi \pi }<510$ MeV and remeasured events with $\cos \theta >0.9995$, comparing adjacent mass bands.^[ref]

For an original dimensionless example, take $m_{\pi }=1$, $\hat{\mathbf{n}}=(0,0,1)$ and ${\mathbf{p}}_{\pm }=(\pm \sqrt{3},0,3)$. Each energy is $\sqrt{13}$; the opening angle is $60$ degrees since $\cos \alpha =(9-3)/(9+3)=1/2$. Yet $\mathbf{P}=(0,0,6)$ points exactly along the beam, and $M_{\pi \pi }^2=4(13)-36=16$. Thus a parent of mass $4$ passes both ideal tests. These chosen numbers illustrate four-momentum conservation, not detector performance.